If your tables share an attribute table field and value, you can join on that field. If you have many fields to add or update export to a new shapefile and delete the fields you don't want. If you only have a field or two to update, add them to the target shapefile and use the calculator to copy from the joined table to the new, then remove the join.
If the tables share no linking fields, you could try a spatial join but that will need checking to be sure it worked properly with odd shaped polygons. If the tables are by default sorted such that each row matches that in the other, you could make a copy of your shapefiles, and then open the dbfs in Excel (older than 2010) or Open Office's spreadsheet, and then copy and paste fields from the source dbf to the target, then delete the fields you don't want. Don't sort and don't touch the shape field.
One other possible option if your license allows, the shapes match very closely and you want to replace the entire table. Use the Polygon to Lines and Polygon to Points (inside polygon checked) tools on both shapefiles. Then use the lines from the target layer and points from the source to build new polygons with new attributes.
